Copula approach to modeling of ARMA and GARCH modelsresiduals

Anna Petričková

FSTA 2012, Liptovský Ján

31.01.2012

Introduction

The state-of-art overview Overview of the ARMA and GARCH models The test of homoscedasticity Copula and autocopula Goodness of fit test for copulas

Application on the hydrological data series Modeling of dependence structure of the ARMA and

GARCH models residuals using autocopulas. Constructing of improved quality models for the

original time series. Conclusion

Linear stochastic models – model ARMA

For example ARMA models

Xt - 1Xt-1 - 2Xt-2 - ... - pXt-p = Zt + 1Zt-1 + 2Zt-2 + ... + qZt-q

where Zt, t = 1, ..., n are i.i.d. process, coefficients 1, ..., p (AR coefficients) and 1, ..., q (MA coefficients) - unknown parameters.

Special cases: If p = 0, we get MA process If q = 0, we get AR process

ARCH and GARCH model

ARCH – AutoRegressive Conditional Heteroscedasticity Let xt is time series in the form

xt = E[xt | t-1] + t

where t-1 is information set containing all relevant information up to time t-1

predictable part E[xt | t-1] is modeled with linear ARMA models

t is unpredictable part with E[t| t-1] = 0, E[t2] = 2.

Model of t in the form

with

where vt ( i.i.d. process with E(t) = 0 and D(t) = 1 ) is called ARCH(m), m is order of the model.

Boundaries for parameters:

2ttt hv

2222

2110 mtmttth

1

0,0,,,0

1

110

m

mm

ARCH and GARCH model

GARCH – Generalized ARCH model. t is time series with E[t

2] = 2 in the form

(1)

where

(2)

and {vt} is white noise process with E(t) = 0 and D(t) = 1.

Time series t generated by (1) and (2) is called generalized ARCH of order p, q, and denote GARCH(p, q).

Boundaries for parameters:

and also

2ttt hv

2211110 qtqtptptt hhh

0,0,,

0,0,,,0

11

110

qq

pp

1)()( 11 qp

McLeod and Li test of Homoscedasticity (1983)

Test statistic

where n is a sample size, rk2 is the squared sample autocorrelation

of squared residual series at lag k and m is moderately large.

When applied to the residuals from an ARMA (p,q) model, the McL test statistic follows distribution asymptotically.)(2 qpm

m

k

k

kn

rnnmMcL

1

22 )ˆ()2()(

2-dimensional copula is a function C: [0, 1]2 [0, 1],

C(0, y) = C(x, 0) = 0, C(1, y) = C(x, 1) = x for all x, y [0, 1] and

C(x1, y1) + C(x2, y2) − C(x1, y2) − C(x2, y1) 0

for all x1, x2, y1, y2 [0, 1], such that x1 x2, y1 y2.

Let F is joint distribution function of 2-dimensional random vectors (X, Y) and FX, FY are marginal distribution functions. Then

F(x, y) = C (FX(x), FY(y)).

Copula C is only one, if X and Y are continuous random variables.

Let Xt is strict stationary time series and k Z+, then autocopulautocopula CX,k is

copula of random vector (Xt, Xt-k).

Copula and autocopula

In our work we used families:

Archimedean class – Gumbel, strict Clayton, Frank, Joe BB1

convex combinations of Archimedean copulas

Extreme Value (EV) Copulas class – Gumbel A, Galambos

Copula and autocopula

Let {(xj, yj), j = 1, …, n } be n modeled 2-dimensional observations, FX, FY their marginal distribution functions and F their joint distribution function.The class of copulas C is correctly specified if there exists 0 so that

White (H. White: Maximum likelihood estimation of misspecified models. Econometrica 50, 1982, pp. 1 – 26) showed that under correct specification of the copula class C holds:

where

00 BA

and c is the density function.

yFxFcyFxFcE

yFxFcE

YXYX

YX

,ln,ln

,ln2

B

A

yF,xFCy,xF YX0

Goodness of fit test for copulas

The testing procedure, which is proposed in A. Prokhorov: A goodness-of-fit test for copulas. MPRA Paper No. 9998, 2008 is based on the empirical distribution functions

and on a consistent estimator of vector of parameters 0.

To introduce the sample versions of A and B put:

n

iiY

n

iiX sy

nsF a sx

nsF

11

11

11

iYiXiYiX

iYiX

yF,xFclnyF,xFcln

yF,xFcln

i

i

B

A 2

n

ii

n

ii

n

n

1

1

1

1

BB

AA

ˆ

,ˆ

Goodness of fit test for copulas

Put:

Under the hypothesis of proper specification the statistics has

asymptotical distribution N(0, V), where V is estimated by

n

iin

ˆ1

1 dD

ii .n

ˆ ddV1

1

Statistics

ii .n

ˆ ddV1

1

is asymptotically as

Dn

2

21 kk

Goodness of fit test for copulas

S. Grønneberg, N. L. Hjort : The copula information criterion. Statistical Research Report , E-print 7, 2008

122

θθn ABTr)θ(LTIC

Takeuchi criterion TIC

Modeling of dependence of residuals of the ARMA and GARCH models with autocopulas

performed using the system MATHEMATICA, version 8

applied the significance level 0.05

from each of the considered time series omitted 12 the most recent values (that were left for purposes of subsequent investigations of the out-of-the-sample forecasting performance of the resulting models)

14 hydrological data series – (monthly) Slovak rivers‘ flows

Application

At first, we have ‘fitted’ these real data series with the ARMAARMA models (seasonally adjusted). We have selected the best model on the basis of the BIC criterion (case 1).

We have fitted autocopulas to the subsequent pairs of the above mentioned residuals of time series. Then we have selected the optimal models that attain the minimum of the TIC criterion. Finally we have applied the best autocopulas instead of the white noise into the original model (case 2).

The residuals of the ARMA models should be homoscedastic, that was checked with McLeod and LiMcLeod and Li test test of homoscedasticity of homoscedasticity.

When homoscedasticity in residuals has been rejected, we have fitted them with ARCH/GARCHARCH/GARCH models (case 3).

Sequence of procedures

Archimedean copulas (AC) 1

convex combinations of AC 13

extreme value copulas 0

The best copulas

Instead of , (which is the strict white noise process with E[et] = 0, D[et] =

e), we have used the autocopulas that we have chosen as the best copulas above (for each real time series).

To compare the quality of the optimal models in all 3 categories we have computed their standard deviations () as well as prediction error RMSE (root mean square error).

te

Improved models

For all 14 (seasonally adjusted) data series fitted with ARMA models McLeod and Li test rejectedrejected homoscedasticity in residuals, so we fitted them with ARCH/GARCH models.

Comparison - description

data

sigma of residuals

ARMA without copula ARMA with copula ARCH/GARCH

Belá - Podbanské 2,40201 2,40913 2,35738

Čierny Váh 1,74083 1,7688 1,86467

Dunaj - Bratislava 0,55422 0,55372 0,86239

Dunajec - Červený Kláštor 1,18901 1,19725 2,37627

Handlovka - Handlová 0,24818 0,25379 0,54604

Hnilec - Jalkovce 0,34886 0,34985 0,71544

Hron - BB 0,13626 0,13072 0,18785

Kysuca - Čadca 0,36173 0,36515 0,82324

Litava - Plastovce 0,10898 0,09281 0,18617

Morava - Moravský Ján 0,59758 0,60372 0,86566

Orava - Drieňová 0,10661 0,09083 0,27903

Poprad - Chmelnica 0,6653 0,66518 1,23505

Topľa - Hanušovce 0,42021 0,42648 0,61885

Torysa - Košické Olšany 0,41571 0,41538 0,79986

Comparison - prediction

data

RMSE

ARMA without copula ARMA with copula ARCH/GARCH

Belá - Podbanské 1,63557 1,66597 2,72753

Čierny Váh 1,25157 1,40243 1,36957

Dunaj - Bratislava 0,37416 0,31338 0,94412

Dunajec - Červený Kláštor 1,82441 1,85133 2,48126

Handlovka - Handlová 0,27643 0,25723 0,35238

Hnilec - Jalkovce 0,43378 0,42289 0,88977

Hron - BB 0,07929 0,07902 0,22236

Kysuca - Čadca 0,37337 0,44147 0,97517

Litava - Plastovce 0,08067 0,08057 0,29661

Morava - Moravský Ján 0,26169 0,35511 0,79046

Orava - Drieňová 0,09936 0,09888 0,26276

Poprad - Chmelnica 0,50249 0,49729 1,1649

Topľa - Hanušovce 0,57753 0,57612 0,56784

Torysa - Košické Olšany 0,57104 0,57075 0,97139

The best descriptive properties belonged to classical ARMA models and ARMA models with copulas, only in 1 case to ARCH/GARCH model.

The best predictive properties had ARMA models with copulas (9) and 5 classical ARMA models. ARCH/GARCH models had the worst RMSE of residuals for all 14 time series.

Improved models

1936 1946 1956 1966 1976 1986 1996 2006time Month 1

2

3

4

5

6

flow m 3s Tory s a K oš ic k é O lš any;

1888 1898 1908 1918 1928 1938 1948 1958 1968 1978 1988 1998 2008time Month

2

4

6

flow m 3s Dunaj B rat is lava

0 2 4 6 8 10 12time Month

1

2

3

4flow m 3s Tory s a K oš ic k é O lš any;

0 2 4 6 8 10 12time Month

1

2

3

4flow m 3s Dunaj B rat is lava

ConclusionsConclusions

We have found out that ARCH/GARCH models are not very suitable for fitting of rivers’ flows data series. Much better attempt was fitting them with classical linear ARMA models and also ARMA models with copulas, where copulas are able to capture wider range of nonlinearity.

In future we also want to describe real time series with non-Archimedean copulas like Gauss, Student copulas, Archimax copulas etc. We also want to use regime-switching model with regimes determined by observable or unobservable variables and compare it with the others.

Thank you for your attention.